Data.Colour.Matrix:determinant from colour-2.3.3, A

Percentage Accurate: 74.5% → 84.7%

Time: 20.0s

Alternatives: 23

Speedup: 0.5×

• 57.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
  (* j (- (* c a) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
  (* j (- (* c a) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(t \cdot b - y \cdot j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (+ (* x (- (* y z) (* t a))) (* b (- (* t i) (* z c))))
          (* j (- (* a c) (* y i))))))
   (if (<= t_1 INFINITY)
     t_1
     (fma c (fma b (- z) (* a j)) (* i (- (* t b) (* y j)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) + (b * ((t * i) - (z * c)))) + (j * ((a * c) - (y * i)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(c, fma(b, -z, (a * j)), (i * ((t * b) - (y * j))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) + Float64(b * Float64(Float64(t * i) - Float64(z * c)))) + Float64(j * Float64(Float64(a * c) - Float64(y * i))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = fma(c, fma(b, Float64(-z), Float64(a * j)), Float64(i * Float64(Float64(t * b) - Float64(y * j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision] + N[(i * N[(N[(t * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 91.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i))))

   1. Initial program 0.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 58.4

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq \infty:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(t \cdot b - y \cdot j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, t \cdot b\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(t \cdot b - y \cdot j\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<=
      (+
       (+ (* x (- (* y z) (* t a))) (* b (- (* t i) (* z c))))
       (* j (- (* a c) (* y i))))
      INFINITY)
   (+
    (fma i (fma j (- y) (* t b)) (* x (fma a (- t) (* y z))))
    (* c (- (* a j) (* z b))))
   (fma c (fma b (- z) (* a j)) (* i (- (* t b) (* y j))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((((x * ((y * z) - (t * a))) + (b * ((t * i) - (z * c)))) + (j * ((a * c) - (y * i)))) <= ((double) INFINITY)) {
		tmp = fma(i, fma(j, -y, (t * b)), (x * fma(a, -t, (y * z)))) + (c * ((a * j) - (z * b)));
	} else {
		tmp = fma(c, fma(b, -z, (a * j)), (i * ((t * b) - (y * j))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) + Float64(b * Float64(Float64(t * i) - Float64(z * c)))) + Float64(j * Float64(Float64(a * c) - Float64(y * i)))) <= Inf)
		tmp = Float64(fma(i, fma(j, Float64(-y), Float64(t * b)), Float64(x * fma(a, Float64(-t), Float64(y * z)))) + Float64(c * Float64(Float64(a * j) - Float64(z * b))));
	else
		tmp = fma(c, fma(b, Float64(-z), Float64(a * j)), Float64(i * Float64(Float64(t * b) - Float64(y * j))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a * (-t) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(a * j), $MachinePrecision] - N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision] + N[(i * N[(N[(t * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 91.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 88.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i))))

   1. Initial program 0.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 58.4

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 58.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(t \cdot i - z \cdot c\right)\right) + j \cdot \left(a \cdot c - y \cdot i\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, t \cdot b\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(a \cdot j - z \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(t \cdot b - y \cdot j\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-115}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) - a \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), -i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma i t (- (* z c))))))
   (if (<= b -1.45e+67)
     t_1
     (if (<= b 9.2e-115)
       (- (* j (- (* a c) (* y i))) (* a (* x t)))
       (if (<= b 7.8e+55)
         (fma c (fma b (- z) (* a j)) (- (* i (* y j))))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(i, t, -(z * c));
	double tmp;
	if (b <= -1.45e+67) {
		tmp = t_1;
	} else if (b <= 9.2e-115) {
		tmp = (j * ((a * c) - (y * i))) - (a * (x * t));
	} else if (b <= 7.8e+55) {
		tmp = fma(c, fma(b, -z, (a * j)), -(i * (y * j)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(i, t, Float64(-Float64(z * c))))
	tmp = 0.0
	if (b <= -1.45e+67)
		tmp = t_1;
	elseif (b <= 9.2e-115)
		tmp = Float64(Float64(j * Float64(Float64(a * c) - Float64(y * i))) - Float64(a * Float64(x * t)));
	elseif (b <= 7.8e+55)
		tmp = fma(c, fma(b, Float64(-z), Float64(a * j)), Float64(-Float64(i * Float64(y * j))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(i * t + (-N[(z * c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.45e+67], t$95$1, If[LessEqual[b, 9.2e-115], N[(N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.8e+55], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision] + (-N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45000000000000012e67 or 7.80000000000000054e55 < b

   1. Initial program 66.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 68.4

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 68.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

       2. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, t, -1 \cdot \left(c \cdot z\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{\mathsf{neg}\left(c \cdot z\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

       6. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(-1 \cdot z\right)}\right) \]

       8. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

       9. neg-lowering-neg.f64 70.5

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-z\right)}\right) \]

   11. Simplified 70.5%

       \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(i, t, c \cdot \left(-z\right)\right)} \]

   if -1.45000000000000012e67 < b < 9.19999999999999938e-115

   1. Initial program 72.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(-1 \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. neg-lowering-neg.f64 66.0

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-x\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{a \cdot \left(t \cdot \left(-x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   if 9.19999999999999938e-115 < b < 7.80000000000000054e55

   1. Initial program 71.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 81.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 78.8

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(j \cdot \left(-1 \cdot y\right)\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(j \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

       6. neg-lowering-neg.f64 69.7

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(j \cdot \color{blue}{\left(-y\right)}\right)\right) \]

   11. Simplified 69.7%

       \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+67}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-115}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) - a \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), -i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right), b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(t \cdot b - y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma a (fma j c (* x (- t))) (* b (- (* t i) (* z c))))))
   (if (<= a -1.9e-24)
     t_1
     (if (<= a 1.8e+95)
       (fma c (fma b (- z) (* a j)) (* i (- (* t b) (* y j))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(a, fma(j, c, (x * -t)), (b * ((t * i) - (z * c))));
	double tmp;
	if (a <= -1.9e-24) {
		tmp = t_1;
	} else if (a <= 1.8e+95) {
		tmp = fma(c, fma(b, -z, (a * j)), (i * ((t * b) - (y * j))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(a, fma(j, c, Float64(x * Float64(-t))), Float64(b * Float64(Float64(t * i) - Float64(z * c))))
	tmp = 0.0
	if (a <= -1.9e-24)
		tmp = t_1;
	elseif (a <= 1.8e+95)
		tmp = fma(c, fma(b, Float64(-z), Float64(a * j)), Float64(i * Float64(Float64(t * b) - Float64(y * j))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.9e-24], t$95$1, If[LessEqual[a, 1.8e+95], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision] + N[(i * N[(N[(t * b), $MachinePrecision] - N[(y * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.90000000000000013e-24 or 1.79999999999999989e95 < a

   1. Initial program 62.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(c \cdot j\right)\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(t \cdot x\right) + c \cdot j, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{c \cdot j + -1 \cdot \left(t \cdot x\right)}, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)}, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

   5. Simplified 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right), b \cdot \left(i \cdot t - c \cdot z\right)\right)} \]

   if -1.90000000000000013e-24 < a < 1.79999999999999989e95

   1. Initial program 75.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 84.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 76.6

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right), b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(t \cdot b - y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right), b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+69}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-101}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) - a \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right), b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -3e+69)
   (* b (fma i t (- (* z c))))
   (if (<= b 2.9e-101)
     (- (* j (- (* a c) (* y i))) (* a (* x t)))
     (fma a (fma j c (* x (- t))) (* b (- (* t i) (* z c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -3e+69) {
		tmp = b * fma(i, t, -(z * c));
	} else if (b <= 2.9e-101) {
		tmp = (j * ((a * c) - (y * i))) - (a * (x * t));
	} else {
		tmp = fma(a, fma(j, c, (x * -t)), (b * ((t * i) - (z * c))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -3e+69)
		tmp = Float64(b * fma(i, t, Float64(-Float64(z * c))));
	elseif (b <= 2.9e-101)
		tmp = Float64(Float64(j * Float64(Float64(a * c) - Float64(y * i))) - Float64(a * Float64(x * t)));
	else
		tmp = fma(a, fma(j, c, Float64(x * Float64(-t))), Float64(b * Float64(Float64(t * i) - Float64(z * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -3e+69], N[(b * N[(i * t + (-N[(z * c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-101], N[(N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.99999999999999983e69

   1. Initial program 63.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 69.7

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 69.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

       2. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, t, -1 \cdot \left(c \cdot z\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{\mathsf{neg}\left(c \cdot z\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

       6. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(-1 \cdot z\right)}\right) \]

       8. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

       9. neg-lowering-neg.f64 71.9

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-z\right)}\right) \]

   11. Simplified 71.9%

       \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(i, t, c \cdot \left(-z\right)\right)} \]

   if -2.99999999999999983e69 < b < 2.9e-101

   1. Initial program 72.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \color{blue}{\left(t \cdot \left(-1 \cdot x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. neg-lowering-neg.f64 65.7

          \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-x\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. Simplified 65.7%

      \[\leadsto \color{blue}{a \cdot \left(t \cdot \left(-x\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   if 2.9e-101 < b

   1. Initial program 70.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(c \cdot j\right)\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right)\right)} + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t \cdot x\right)\right)} + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right)\right)} + a \cdot \left(c \cdot j\right)\right) + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} + \left(\mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(t \cdot x\right) + c \cdot j, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{c \cdot j + -1 \cdot \left(t \cdot x\right)}, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)}, \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right), \mathsf{neg}\left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(\mathsf{neg}\left(x\right)\right)\right), b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)}\right) \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right), b \cdot \left(i \cdot t - c \cdot z\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+69}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-101}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right) - a \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right), b \cdot \left(t \cdot i - z \cdot c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), -i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma a (- x) (* b i)))))
   (if (<= t -1.7e+21)
     t_1
     (if (<= t 1.55e+81)
       (fma c (fma b (- z) (* a j)) (- (* i (* y j))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * fma(a, -x, (b * i));
	double tmp;
	if (t <= -1.7e+21) {
		tmp = t_1;
	} else if (t <= 1.55e+81) {
		tmp = fma(c, fma(b, -z, (a * j)), -(i * (y * j)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(a, Float64(-x), Float64(b * i)))
	tmp = 0.0
	if (t <= -1.7e+21)
		tmp = t_1;
	elseif (t <= 1.55e+81)
		tmp = fma(c, fma(b, Float64(-z), Float64(a * j)), Float64(-Float64(i * Float64(y * j))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(a * (-x) + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.7e+21], t$95$1, If[LessEqual[t, 1.55e+81], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision] + (-N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e21 or 1.55e81 < t

   1. Initial program 56.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto t \cdot \color{blue}{\left(-1 \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto t \cdot \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto t \cdot \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto t \cdot \left(a \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto t \cdot \left(a \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto t \cdot \left(a \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto t \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto t \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto t \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto t \cdot \mathsf{fma}\left(a, \mathsf{neg}\left(x\right), \color{blue}{i \cdot b}\right) \]

      12. *-lowering-*.f64 72.4

          \[\leadsto t \cdot \mathsf{fma}\left(a, -x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 72.4%

      \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(a, -x, i \cdot b\right)} \]

   if -1.7e21 < t < 1.55e81

   1. Initial program 79.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 80.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 72.2

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)}\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(j \cdot \left(-1 \cdot y\right)\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(j \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

       6. neg-lowering-neg.f64 61.3

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(j \cdot \color{blue}{\left(-y\right)}\right)\right) \]

   11. Simplified 61.3%

       \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+21}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), -i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(a, -x, b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ t_2 := i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{if}\;i \leq -2 \cdot 10^{+46}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;i \leq -1.36 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-270}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma j c (* x (- t))))) (t_2 (* i (fma j (- y) (* t b)))))
   (if (<= i -2e+46)
     t_2
     (if (<= i -1.36e-49)
       t_1
       (if (<= i 3e-270)
         (* z (fma c (- b) (* x y)))
         (if (<= i 3e-92) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(j, c, (x * -t));
	double t_2 = i * fma(j, -y, (t * b));
	double tmp;
	if (i <= -2e+46) {
		tmp = t_2;
	} else if (i <= -1.36e-49) {
		tmp = t_1;
	} else if (i <= 3e-270) {
		tmp = z * fma(c, -b, (x * y));
	} else if (i <= 3e-92) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(j, c, Float64(x * Float64(-t))))
	t_2 = Float64(i * fma(j, Float64(-y), Float64(t * b)))
	tmp = 0.0
	if (i <= -2e+46)
		tmp = t_2;
	elseif (i <= -1.36e-49)
		tmp = t_1;
	elseif (i <= 3e-270)
		tmp = Float64(z * fma(c, Float64(-b), Float64(x * y)));
	elseif (i <= 3e-92)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -2e+46], t$95$2, If[LessEqual[i, -1.36e-49], t$95$1, If[LessEqual[i, 3e-270], N[(z * N[(c * (-b) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 3e-92], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if i < -2e46 or 3.00000000000000013e-92 < i

   1. Initial program 64.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 67.7

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 67.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   if -2e46 < i < -1.36000000000000006e-49 or 3.00000000000000013e-270 < i < 3.00000000000000013e-92

   1. Initial program 73.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 60.4

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   if -1.36000000000000006e-49 < i < 3.00000000000000013e-270

   1. Initial program 82.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      10. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(b\right), \color{blue}{y \cdot x}\right) \]

      11. *-lowering-*.f64 65.5

          \[\leadsto z \cdot \mathsf{fma}\left(c, -b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, -b, y \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2 \cdot 10^{+46}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{elif}\;i \leq -1.36 \cdot 10^{-49}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-270}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(c, -b, x \cdot y\right)\\ \mathbf{elif}\;i \leq 3 \cdot 10^{-92}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{if}\;i \leq -1.18 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \left(y \cdot \mathsf{fma}\left(t, \frac{a}{-y}, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* t b)))))
   (if (<= i -1.18e+47)
     t_1
     (if (<= i 2.1e-78) (* x (* y (fma t (/ a (- y)) z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, -y, (t * b));
	double tmp;
	if (i <= -1.18e+47) {
		tmp = t_1;
	} else if (i <= 2.1e-78) {
		tmp = x * (y * fma(t, (a / -y), z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(t * b)))
	tmp = 0.0
	if (i <= -1.18e+47)
		tmp = t_1;
	elseif (i <= 2.1e-78)
		tmp = Float64(x * Float64(y * fma(t, Float64(a / Float64(-y)), z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.18e+47], t$95$1, If[LessEqual[i, 2.1e-78], N[(x * N[(y * N[(t * N[(a / (-y)), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.18e47 or 2.1000000000000001e-78 < i

   1. Initial program 64.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 68.1

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 68.1%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   if -1.18e47 < i < 2.1000000000000001e-78

   1. Initial program 77.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t\right)\right)} + y \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot t\right)} + y \cdot z\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot t, y \cdot z\right)} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      9. *-lowering-*.f64 54.3

         \[\leadsto x \cdot \mathsf{fma}\left(a, -t, \color{blue}{y \cdot z}\right) \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)} \]

   6. Taylor expanded in y around -inf

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \left(y \cdot \left(-1 \cdot z + \frac{a \cdot t}{y}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(-1 \cdot z + \frac{a \cdot t}{y}\right)\right)\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(-1 \cdot z + \frac{a \cdot t}{y}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot \left(-1 \cdot z + \frac{a \cdot t}{y}\right)}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(y \cdot \color{blue}{\left(\frac{a \cdot t}{y} + -1 \cdot z\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(y \cdot \left(\frac{\color{blue}{t \cdot a}}{y} + -1 \cdot z\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(y \cdot \left(\color{blue}{t \cdot \frac{a}{y}} + -1 \cdot z\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(y \cdot \color{blue}{\mathsf{fma}\left(t, \frac{a}{y}, -1 \cdot z\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(y \cdot \mathsf{fma}\left(t, \color{blue}{\frac{a}{y}}, -1 \cdot z\right)\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(y \cdot \mathsf{fma}\left(t, \frac{a}{y}, \color{blue}{\mathsf{neg}\left(z\right)}\right)\right)\right) \]

      10. neg-lowering-neg.f64 57.1

          \[\leadsto x \cdot \left(-y \cdot \mathsf{fma}\left(t, \frac{a}{y}, \color{blue}{-z}\right)\right) \]

   8. Simplified 57.1%

      \[\leadsto x \cdot \color{blue}{\left(-y \cdot \mathsf{fma}\left(t, \frac{a}{y}, -z\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot \left(\frac{a \cdot t}{y} - z\right)\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto -1 \cdot \color{blue}{\left(\left(y \cdot \left(\frac{a \cdot t}{y} - z\right)\right) \cdot x\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(\frac{a \cdot t}{y} - z\right)\right)\right) \cdot x} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot y\right) \cdot \left(\frac{a \cdot t}{y} - z\right)\right)} \cdot x \]

       4. sub-neg N/A

          \[\leadsto \left(\left(-1 \cdot y\right) \cdot \color{blue}{\left(\frac{a \cdot t}{y} + \left(\mathsf{neg}\left(z\right)\right)\right)}\right) \cdot x \]

       5. mul-1-neg N/A

          \[\leadsto \left(\left(-1 \cdot y\right) \cdot \left(\frac{a \cdot t}{y} + \color{blue}{-1 \cdot z}\right)\right) \cdot x \]

       6. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(\left(-1 \cdot y\right) \cdot \frac{a \cdot t}{y} + \left(-1 \cdot y\right) \cdot \left(-1 \cdot z\right)\right)} \cdot x \]

       7. mul-1-neg N/A

          \[\leadsto \left(\left(-1 \cdot y\right) \cdot \frac{a \cdot t}{y} + \left(-1 \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \cdot x \]

       8. distribute-rgt-neg-in N/A

          \[\leadsto \left(\left(-1 \cdot y\right) \cdot \frac{a \cdot t}{y} + \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot y\right) \cdot z\right)\right)}\right) \cdot x \]

       9. associate-*r* N/A

          \[\leadsto \left(\left(-1 \cdot y\right) \cdot \frac{a \cdot t}{y} + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(y \cdot z\right)}\right)\right)\right) \cdot x \]

       10. mul-1-neg N/A

           \[\leadsto \left(\left(-1 \cdot y\right) \cdot \frac{a \cdot t}{y} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)}\right)\right)\right) \cdot x \]

       11. remove-double-neg N/A

           \[\leadsto \left(\left(-1 \cdot y\right) \cdot \frac{a \cdot t}{y} + \color{blue}{y \cdot z}\right) \cdot x \]

       12. *-commutative N/A

           \[\leadsto \left(\color{blue}{\left(y \cdot -1\right)} \cdot \frac{a \cdot t}{y} + y \cdot z\right) \cdot x \]

       13. associate-*r* N/A

           \[\leadsto \left(\color{blue}{y \cdot \left(-1 \cdot \frac{a \cdot t}{y}\right)} + y \cdot z\right) \cdot x \]

       14. distribute-lft-in N/A

           \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \frac{a \cdot t}{y} + z\right)\right)} \cdot x \]

       15. +-commutative N/A

           \[\leadsto \left(y \cdot \color{blue}{\left(z + -1 \cdot \frac{a \cdot t}{y}\right)}\right) \cdot x \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{\left(y \cdot \left(z + -1 \cdot \frac{a \cdot t}{y}\right)\right) \cdot x} \]

   11. Simplified 57.1%

       \[\leadsto \color{blue}{\left(y \cdot \mathsf{fma}\left(t, \frac{a}{-y}, z\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.18 \cdot 10^{+47}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{elif}\;i \leq 2.1 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \left(y \cdot \mathsf{fma}\left(t, \frac{a}{-y}, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -5.3 \cdot 10^{+144}:\\ \;\;\;\;-i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;j \leq -1.18 \cdot 10^{-240}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{+20}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(j \cdot \left(-i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -5.3e+144)
   (- (* i (* y j)))
   (if (<= j -1.18e-240)
     (* a (fma j c (* x (- t))))
     (if (<= j 7.8e+20) (* i (* t b)) (* y (* j (- i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -5.3e+144) {
		tmp = -(i * (y * j));
	} else if (j <= -1.18e-240) {
		tmp = a * fma(j, c, (x * -t));
	} else if (j <= 7.8e+20) {
		tmp = i * (t * b);
	} else {
		tmp = y * (j * -i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -5.3e+144)
		tmp = Float64(-Float64(i * Float64(y * j)));
	elseif (j <= -1.18e-240)
		tmp = Float64(a * fma(j, c, Float64(x * Float64(-t))));
	elseif (j <= 7.8e+20)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = Float64(y * Float64(j * Float64(-i)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -5.3e+144], (-N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), If[LessEqual[j, -1.18e-240], N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 7.8e+20], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], N[(y * N[(j * (-i)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -5.2999999999999997e144

   1. Initial program 61.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 64.7

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-1 \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      6. neg-lowering-neg.f64 60.5

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-y\right)}\right) \]

   8. Simplified 60.5%

      \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)} \]

   if -5.2999999999999997e144 < j < -1.18e-240

   1. Initial program 75.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 49.7

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 49.7%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   if -1.18e-240 < j < 7.8e20

   1. Initial program 69.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 50.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 50.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 41.7

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Simplified 41.7%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   if 7.8e20 < j

   1. Initial program 69.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 55.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 55.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-1 \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      6. neg-lowering-neg.f64 43.5

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-y\right)}\right) \]

   8. Simplified 43.5%

      \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right) \cdot i} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{j \cdot \left(\left(\mathsf{neg}\left(y\right)\right) \cdot i\right)} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto j \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot i\right)\right)} \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto j \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(i\right)\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \cdot y} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \cdot y} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(i\right)\right)\right)} \cdot y \]

      9. neg-lowering-neg.f64 51.0

         \[\leadsto \left(j \cdot \color{blue}{\left(-i\right)}\right) \cdot y \]

   10. Applied egg-rr 51.0%

       \[\leadsto \color{blue}{\left(j \cdot \left(-i\right)\right) \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -5.3 \cdot 10^{+144}:\\ \;\;\;\;-i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;j \leq -1.18 \cdot 10^{-240}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;j \leq 7.8 \cdot 10^{+20}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(j \cdot \left(-i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.8 \cdot 10^{+144}:\\ \;\;\;\;-i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;j \leq -0.012:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{+22}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(j \cdot \left(-i\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -1.8e+144)
   (- (* i (* y j)))
   (if (<= j -0.012)
     (* a (* c j))
     (if (<= j 9.5e+22) (* i (* t b)) (* y (* j (- i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1.8e+144) {
		tmp = -(i * (y * j));
	} else if (j <= -0.012) {
		tmp = a * (c * j);
	} else if (j <= 9.5e+22) {
		tmp = i * (t * b);
	} else {
		tmp = y * (j * -i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (j <= (-1.8d+144)) then
        tmp = -(i * (y * j))
    else if (j <= (-0.012d0)) then
        tmp = a * (c * j)
    else if (j <= 9.5d+22) then
        tmp = i * (t * b)
    else
        tmp = y * (j * -i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -1.8e+144) {
		tmp = -(i * (y * j));
	} else if (j <= -0.012) {
		tmp = a * (c * j);
	} else if (j <= 9.5e+22) {
		tmp = i * (t * b);
	} else {
		tmp = y * (j * -i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if j <= -1.8e+144:
		tmp = -(i * (y * j))
	elif j <= -0.012:
		tmp = a * (c * j)
	elif j <= 9.5e+22:
		tmp = i * (t * b)
	else:
		tmp = y * (j * -i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -1.8e+144)
		tmp = Float64(-Float64(i * Float64(y * j)));
	elseif (j <= -0.012)
		tmp = Float64(a * Float64(c * j));
	elseif (j <= 9.5e+22)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = Float64(y * Float64(j * Float64(-i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (j <= -1.8e+144)
		tmp = -(i * (y * j));
	elseif (j <= -0.012)
		tmp = a * (c * j);
	elseif (j <= 9.5e+22)
		tmp = i * (t * b);
	else
		tmp = y * (j * -i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -1.8e+144], (-N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision]), If[LessEqual[j, -0.012], N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 9.5e+22], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], N[(y * N[(j * (-i)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if j < -1.7999999999999999e144

   1. Initial program 61.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 64.7

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-1 \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      6. neg-lowering-neg.f64 60.5

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-y\right)}\right) \]

   8. Simplified 60.5%

      \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)} \]

   if -1.7999999999999999e144 < j < -0.012

   1. Initial program 69.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 69.5

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 69.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

      2. *-lowering-*.f64 54.9

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

   8. Simplified 54.9%

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

   if -0.012 < j < 9.49999999999999937e22

   1. Initial program 72.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 44.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 44.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 36.7

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Simplified 36.7%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   if 9.49999999999999937e22 < j

   1. Initial program 69.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 55.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 55.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-1 \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      6. neg-lowering-neg.f64 43.5

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-y\right)}\right) \]

   8. Simplified 43.5%

      \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right) \cdot i} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{j \cdot \left(\left(\mathsf{neg}\left(y\right)\right) \cdot i\right)} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto j \cdot \color{blue}{\left(\mathsf{neg}\left(y \cdot i\right)\right)} \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto j \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(i\right)\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(\left(\mathsf{neg}\left(i\right)\right) \cdot y\right)} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \cdot y} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(i\right)\right)\right) \cdot y} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(i\right)\right)\right)} \cdot y \]

      9. neg-lowering-neg.f64 51.0

         \[\leadsto \left(j \cdot \color{blue}{\left(-i\right)}\right) \cdot y \]

   10. Applied egg-rr 51.0%

       \[\leadsto \color{blue}{\left(j \cdot \left(-i\right)\right) \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.8 \cdot 10^{+144}:\\ \;\;\;\;-i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;j \leq -0.012:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{+22}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(j \cdot \left(-i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -i \cdot \left(y \cdot j\right)\\ \mathbf{if}\;j \leq -3.3 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -0.000225:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* i (* y j)))))
   (if (<= j -3.3e+143)
     t_1
     (if (<= j -0.000225)
       (* a (* c j))
       (if (<= j 8.5e+22) (* i (* t b)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -(i * (y * j));
	double tmp;
	if (j <= -3.3e+143) {
		tmp = t_1;
	} else if (j <= -0.000225) {
		tmp = a * (c * j);
	} else if (j <= 8.5e+22) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -(i * (y * j))
    if (j <= (-3.3d+143)) then
        tmp = t_1
    else if (j <= (-0.000225d0)) then
        tmp = a * (c * j)
    else if (j <= 8.5d+22) then
        tmp = i * (t * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -(i * (y * j));
	double tmp;
	if (j <= -3.3e+143) {
		tmp = t_1;
	} else if (j <= -0.000225) {
		tmp = a * (c * j);
	} else if (j <= 8.5e+22) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = -(i * (y * j))
	tmp = 0
	if j <= -3.3e+143:
		tmp = t_1
	elif j <= -0.000225:
		tmp = a * (c * j)
	elif j <= 8.5e+22:
		tmp = i * (t * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(-Float64(i * Float64(y * j)))
	tmp = 0.0
	if (j <= -3.3e+143)
		tmp = t_1;
	elseif (j <= -0.000225)
		tmp = Float64(a * Float64(c * j));
	elseif (j <= 8.5e+22)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = -(i * (y * j));
	tmp = 0.0;
	if (j <= -3.3e+143)
		tmp = t_1;
	elseif (j <= -0.000225)
		tmp = a * (c * j);
	elseif (j <= 8.5e+22)
		tmp = i * (t * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = (-N[(i * N[(y * j), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[j, -3.3e+143], t$95$1, If[LessEqual[j, -0.000225], N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 8.5e+22], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -3.3e143 or 8.49999999999999979e22 < j

   1. Initial program 66.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 59.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-1 \cdot y\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      6. neg-lowering-neg.f64 50.7

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-y\right)}\right) \]

   8. Simplified 50.7%

      \[\leadsto i \cdot \color{blue}{\left(j \cdot \left(-y\right)\right)} \]

   if -3.3e143 < j < -2.2499999999999999e-4

   1. Initial program 69.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 69.5

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 69.5%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

      2. *-lowering-*.f64 54.9

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

   8. Simplified 54.9%

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

   if -2.2499999999999999e-4 < j < 8.49999999999999979e22

   1. Initial program 72.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 44.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 44.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 36.7

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Simplified 36.7%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -3.3 \cdot 10^{+143}:\\ \;\;\;\;-i \cdot \left(y \cdot j\right)\\ \mathbf{elif}\;j \leq -0.000225:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;j \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-i \cdot \left(y \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{if}\;i \leq -1.62 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;i \leq 4.6 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (fma j (- y) (* t b)))))
   (if (<= i -1.62e+46)
     t_1
     (if (<= i 4.6e-83) (* x (fma a (- t) (* y z))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * fma(j, -y, (t * b));
	double tmp;
	if (i <= -1.62e+46) {
		tmp = t_1;
	} else if (i <= 4.6e-83) {
		tmp = x * fma(a, -t, (y * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * fma(j, Float64(-y), Float64(t * b)))
	tmp = 0.0
	if (i <= -1.62e+46)
		tmp = t_1;
	elseif (i <= 4.6e-83)
		tmp = Float64(x * fma(a, Float64(-t), Float64(y * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, -1.62e+46], t$95$1, If[LessEqual[i, 4.6e-83], N[(x * N[(a * (-t) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if i < -1.6200000000000001e46 or 4.59999999999999979e-83 < i

   1. Initial program 64.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 68.1

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 68.1%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   if -1.6200000000000001e46 < i < 4.59999999999999979e-83

   1. Initial program 77.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t\right)\right)} + y \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot t\right)} + y \cdot z\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot t, y \cdot z\right)} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      9. *-lowering-*.f64 54.3

         \[\leadsto x \cdot \mathsf{fma}\left(a, -t, \color{blue}{y \cdot z}\right) \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.62 \cdot 10^{+46}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{elif}\;i \leq 4.6 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, a, y \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma i t (- (* z c))))))
   (if (<= b -4.2e+39)
     t_1
     (if (<= b 7.6e-14) (* j (fma c a (* y (- i)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(i, t, -(z * c));
	double tmp;
	if (b <= -4.2e+39) {
		tmp = t_1;
	} else if (b <= 7.6e-14) {
		tmp = j * fma(c, a, (y * -i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(i, t, Float64(-Float64(z * c))))
	tmp = 0.0
	if (b <= -4.2e+39)
		tmp = t_1;
	elseif (b <= 7.6e-14)
		tmp = Float64(j * fma(c, a, Float64(y * Float64(-i))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(i * t + (-N[(z * c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.2e+39], t$95$1, If[LessEqual[b, 7.6e-14], N[(j * N[(c * a + N[(y * (-i)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.1999999999999997e39 or 7.6000000000000004e-14 < b

   1. Initial program 67.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 66.6

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

       2. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, t, -1 \cdot \left(c \cdot z\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{\mathsf{neg}\left(c \cdot z\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

       6. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(-1 \cdot z\right)}\right) \]

       8. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

       9. neg-lowering-neg.f64 65.6

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-z\right)}\right) \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(i, t, c \cdot \left(-z\right)\right)} \]

   if -4.1999999999999997e39 < b < 7.6000000000000004e-14

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + a \cdot c\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(a \cdot c + -1 \cdot \left(i \cdot y\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto j \cdot \left(a \cdot c + \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(a \cdot c - i \cdot y\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]

      5. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(a \cdot c + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto j \cdot \left(\color{blue}{c \cdot a} + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot a + \color{blue}{-1 \cdot \left(i \cdot y\right)}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, a, -1 \cdot \left(i \cdot y\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, a, \color{blue}{\mathsf{neg}\left(i \cdot y\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto j \cdot \mathsf{fma}\left(c, a, \mathsf{neg}\left(\color{blue}{y \cdot i}\right)\right) \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto j \cdot \mathsf{fma}\left(c, a, \color{blue}{y \cdot \left(\mathsf{neg}\left(i\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto j \cdot \mathsf{fma}\left(c, a, y \cdot \color{blue}{\left(-1 \cdot i\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto j \cdot \mathsf{fma}\left(c, a, \color{blue}{y \cdot \left(-1 \cdot i\right)}\right) \]

      14. mul-1-neg N/A

          \[\leadsto j \cdot \mathsf{fma}\left(c, a, y \cdot \color{blue}{\left(\mathsf{neg}\left(i\right)\right)}\right) \]

      15. neg-lowering-neg.f64 57.2

          \[\leadsto j \cdot \mathsf{fma}\left(c, a, y \cdot \color{blue}{\left(-i\right)}\right) \]

   8. Simplified 57.2%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, a, y \cdot \left(-i\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+39}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, a, y \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{if}\;b \leq -1.08 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-16}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (fma i t (- (* z c))))))
   (if (<= b -1.08e+39)
     t_1
     (if (<= b 7.5e-16) (* j (- (* a c) (* y i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * fma(i, t, -(z * c));
	double tmp;
	if (b <= -1.08e+39) {
		tmp = t_1;
	} else if (b <= 7.5e-16) {
		tmp = j * ((a * c) - (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * fma(i, t, Float64(-Float64(z * c))))
	tmp = 0.0
	if (b <= -1.08e+39)
		tmp = t_1;
	elseif (b <= 7.5e-16)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(i * t + (-N[(z * c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.08e+39], t$95$1, If[LessEqual[b, 7.5e-16], N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.07999999999999998e39 or 7.5e-16 < b

   1. Initial program 67.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 66.6

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

       2. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, t, -1 \cdot \left(c \cdot z\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{\mathsf{neg}\left(c \cdot z\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

       6. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(-1 \cdot z\right)}\right) \]

       8. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

       9. neg-lowering-neg.f64 65.6

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-z\right)}\right) \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(i, t, c \cdot \left(-z\right)\right)} \]

   if -1.07999999999999998e39 < b < 7.5e-16

   1. Initial program 72.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(a \cdot c - i \cdot y\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \left(\color{blue}{a \cdot c} - i \cdot y\right) \]

      4. *-lowering-*.f64 57.2

         \[\leadsto j \cdot \left(a \cdot c - \color{blue}{i \cdot y}\right) \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.08 \cdot 10^{+39}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-16}:\\ \;\;\;\;j \cdot \left(a \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1.45e-19)
   (* c (fma b (- z) (* a j)))
   (if (<= c 1.7e+85)
     (* i (fma j (- y) (* t b)))
     (* b (fma i t (- (* z c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1.45e-19) {
		tmp = c * fma(b, -z, (a * j));
	} else if (c <= 1.7e+85) {
		tmp = i * fma(j, -y, (t * b));
	} else {
		tmp = b * fma(i, t, -(z * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1.45e-19)
		tmp = Float64(c * fma(b, Float64(-z), Float64(a * j)));
	elseif (c <= 1.7e+85)
		tmp = Float64(i * fma(j, Float64(-y), Float64(t * b)));
	else
		tmp = Float64(b * fma(i, t, Float64(-Float64(z * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1.45e-19], N[(c * N[(b * (-z) + N[(a * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e+85], N[(i * N[(j * (-y) + N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(i * t + (-N[(z * c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.45e-19

   1. Initial program 61.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j\right) \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)} \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right) \]

      9. *-lowering-*.f64 71.9

         \[\leadsto c \cdot \mathsf{fma}\left(b, -z, \color{blue}{a \cdot j}\right) \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)} \]

   if -1.45e-19 < c < 1.7000000000000002e85

   1. Initial program 76.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 57.1

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   if 1.7000000000000002e85 < c

   1. Initial program 54.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 69.1

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 69.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

       2. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, t, -1 \cdot \left(c \cdot z\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{\mathsf{neg}\left(c \cdot z\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

       6. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(-1 \cdot z\right)}\right) \]

       8. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

       9. neg-lowering-neg.f64 59.6

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-z\right)}\right) \]

   11. Simplified 59.6%

       \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(i, t, c \cdot \left(-z\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, -z, a \cdot j\right)\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;i \cdot \mathsf{fma}\left(j, -y, t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+37}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma j c (* x (- t))))))
   (if (<= a -1.1e+143)
     t_1
     (if (<= a 4.8e+37) (* b (fma i t (- (* z c)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(j, c, (x * -t));
	double tmp;
	if (a <= -1.1e+143) {
		tmp = t_1;
	} else if (a <= 4.8e+37) {
		tmp = b * fma(i, t, -(z * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(j, c, Float64(x * Float64(-t))))
	tmp = 0.0
	if (a <= -1.1e+143)
		tmp = t_1;
	elseif (a <= 4.8e+37)
		tmp = Float64(b * fma(i, t, Float64(-Float64(z * c))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.1e+143], t$95$1, If[LessEqual[a, 4.8e+37], N[(b * N[(i * t + (-N[(z * c), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.10000000000000007e143 or 4.8e37 < a

   1. Initial program 61.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 65.2

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   if -1.10000000000000007e143 < a < 4.8e37

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, a \cdot j - b \cdot z, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{a \cdot j + \left(\mathsf{neg}\left(b \cdot z\right)\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right) + a \cdot j}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, b \cdot \color{blue}{\left(-1 \cdot z\right)} + a \cdot j, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, a \cdot j\right)}, i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, a \cdot j\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), \color{blue}{a \cdot j}\right), i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right)}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(b \cdot t + \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)}\right)\right) \]

      13. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \color{blue}{\left(b \cdot t - j \cdot y\right)}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, \mathsf{neg}\left(z\right), a \cdot j\right), i \cdot \left(\color{blue}{b \cdot t} - j \cdot y\right)\right) \]

      16. *-lowering-*.f64 72.4

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - \color{blue}{j \cdot y}\right)\right) \]

   8. Simplified 72.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(b, -z, a \cdot j\right), i \cdot \left(b \cdot t - j \cdot y\right)\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

       2. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t + -1 \cdot \left(c \cdot z\right)\right)} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(i, t, -1 \cdot \left(c \cdot z\right)\right)} \]

       4. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{\mathsf{neg}\left(c \cdot z\right)}\right) \]

       5. distribute-rgt-neg-in N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)}\right) \]

       6. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-1 \cdot z\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, \color{blue}{c \cdot \left(-1 \cdot z\right)}\right) \]

       8. mul-1-neg N/A

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right) \]

       9. neg-lowering-neg.f64 52.3

          \[\leadsto b \cdot \mathsf{fma}\left(i, t, c \cdot \color{blue}{\left(-z\right)}\right) \]

   11. Simplified 52.3%

       \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(i, t, c \cdot \left(-z\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+143}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+37}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(i, t, -z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{if}\;a \leq -8 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma j c (* x (- t))))))
   (if (<= a -8e+146)
     t_1
     (if (<= a 1.05e+38) (* b (fma c (- z) (* t i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(j, c, (x * -t));
	double tmp;
	if (a <= -8e+146) {
		tmp = t_1;
	} else if (a <= 1.05e+38) {
		tmp = b * fma(c, -z, (t * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(j, c, Float64(x * Float64(-t))))
	tmp = 0.0
	if (a <= -8e+146)
		tmp = t_1;
	elseif (a <= 1.05e+38)
		tmp = Float64(b * fma(c, Float64(-z), Float64(t * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -8e+146], t$95$1, If[LessEqual[a, 1.05e+38], N[(b * N[(c * (-z) + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.99999999999999947e146 or 1.05e38 < a

   1. Initial program 61.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 65.2

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   if -7.99999999999999947e146 < a < 1.05e38

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\left(a \cdot \left(c \cdot j\right) + \left(i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right)\right) - b \cdot \left(c \cdot z\right)} \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, -y, b \cdot t\right), x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)\right) + c \cdot \left(j \cdot a - b \cdot z\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + i \cdot t\right)} \]

      2. mul-1-neg N/A

         \[\leadsto b \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)} + i \cdot t\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto b \cdot \left(\color{blue}{c \cdot \left(\mathsf{neg}\left(z\right)\right)} + i \cdot t\right) \]

      4. mul-1-neg N/A

         \[\leadsto b \cdot \left(c \cdot \color{blue}{\left(-1 \cdot z\right)} + i \cdot t\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot z, i \cdot t\right)} \]

      6. mul-1-neg N/A

         \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, i \cdot t\right) \]

      7. neg-lowering-neg.f64 N/A

         \[\leadsto b \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(z\right)}, i \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto b \cdot \mathsf{fma}\left(c, \mathsf{neg}\left(z\right), \color{blue}{t \cdot i}\right) \]

      9. *-lowering-*.f64 51.7

         \[\leadsto b \cdot \mathsf{fma}\left(c, -z, \color{blue}{t \cdot i}\right) \]

   8. Simplified 51.7%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+146}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+38}:\\ \;\;\;\;b \cdot \mathsf{fma}\left(c, -z, t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+37}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma j c (* x (- t))))))
   (if (<= a -2.1e+143)
     t_1
     (if (<= a 4.3e+37) (* b (- (* t i) (* z c))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(j, c, (x * -t));
	double tmp;
	if (a <= -2.1e+143) {
		tmp = t_1;
	} else if (a <= 4.3e+37) {
		tmp = b * ((t * i) - (z * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(j, c, Float64(x * Float64(-t))))
	tmp = 0.0
	if (a <= -2.1e+143)
		tmp = t_1;
	elseif (a <= 4.3e+37)
		tmp = Float64(b * Float64(Float64(t * i) - Float64(z * c)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(j * c + N[(x * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.1e+143], t$95$1, If[LessEqual[a, 4.3e+37], N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.09999999999999988e143 or 4.2999999999999997e37 < a

   1. Initial program 61.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 65.2

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 65.2%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   if -2.09999999999999988e143 < a < 4.2999999999999997e37

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot z + i \cdot t\right)} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto b \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)} + i \cdot t\right) \]

      4. remove-double-neg N/A

         \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \]

      6. sub-neg N/A

         \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z - i \cdot t\right)}\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z - i \cdot t\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\mathsf{neg}\left(\left(c \cdot z - i \cdot t\right)\right)\right)} \]

      10. sub-neg N/A

          \[\leadsto b \cdot \left(\mathsf{neg}\left(\color{blue}{\left(c \cdot z + \left(\mathsf{neg}\left(i \cdot t\right)\right)\right)}\right)\right) \]

      11. distribute-neg-in N/A

          \[\leadsto b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(i \cdot t\right)\right)\right)\right)\right)} \]

      12. remove-double-neg N/A

          \[\leadsto b \cdot \left(\left(\mathsf{neg}\left(c \cdot z\right)\right) + \color{blue}{i \cdot t}\right) \]

      13. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t + \left(\mathsf{neg}\left(c \cdot z\right)\right)\right)} \]

      14. sub-neg N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t - c \cdot z\right)} \]

      15. --lowering--.f64 N/A

          \[\leadsto b \cdot \color{blue}{\left(i \cdot t - c \cdot z\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \left(\color{blue}{i \cdot t} - c \cdot z\right) \]

      17. *-lowering-*.f64 51.7

          \[\leadsto b \cdot \left(i \cdot t - \color{blue}{c \cdot z}\right) \]

   5. Simplified 51.7%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+143}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{+37}:\\ \;\;\;\;b \cdot \left(t \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(j, c, x \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.08 \cdot 10^{+40}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;i \leq 1.26 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -1.08e+40)
   (* b (* t i))
   (if (<= i 1.26e-179) (* x (* y z)) (* t (* b i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.08e+40) {
		tmp = b * (t * i);
	} else if (i <= 1.26e-179) {
		tmp = x * (y * z);
	} else {
		tmp = t * (b * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (i <= (-1.08d+40)) then
        tmp = b * (t * i)
    else if (i <= 1.26d-179) then
        tmp = x * (y * z)
    else
        tmp = t * (b * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -1.08e+40) {
		tmp = b * (t * i);
	} else if (i <= 1.26e-179) {
		tmp = x * (y * z);
	} else {
		tmp = t * (b * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if i <= -1.08e+40:
		tmp = b * (t * i)
	elif i <= 1.26e-179:
		tmp = x * (y * z)
	else:
		tmp = t * (b * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -1.08e+40)
		tmp = Float64(b * Float64(t * i));
	elseif (i <= 1.26e-179)
		tmp = Float64(x * Float64(y * z));
	else
		tmp = Float64(t * Float64(b * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (i <= -1.08e+40)
		tmp = b * (t * i);
	elseif (i <= 1.26e-179)
		tmp = x * (y * z);
	else
		tmp = t * (b * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -1.08e+40], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.26e-179], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(t * N[(b * i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -1.08000000000000001e40

   1. Initial program 63.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 63.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 63.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

      3. *-lowering-*.f64 43.1

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

   8. Simplified 43.1%

      \[\leadsto \color{blue}{b \cdot \left(t \cdot i\right)} \]

   if -1.08000000000000001e40 < i < 1.2599999999999999e-179

   1. Initial program 79.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t\right)\right)} + y \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot t\right)} + y \cdot z\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot t, y \cdot z\right)} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      9. *-lowering-*.f64 56.2

         \[\leadsto x \cdot \mathsf{fma}\left(a, -t, \color{blue}{y \cdot z}\right) \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      3. *-lowering-*.f64 35.6

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Simplified 35.6%

      \[\leadsto \color{blue}{x \cdot \left(z \cdot y\right)} \]

   if 1.2599999999999999e-179 < i

   1. Initial program 64.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 65.7

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 65.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

      3. *-lowering-*.f64 41.5

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

   8. Simplified 41.5%

      \[\leadsto \color{blue}{b \cdot \left(t \cdot i\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(b \cdot i\right) \cdot t} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot i\right) \cdot t} \]

      4. *-lowering-*.f64 41.5

         \[\leadsto \color{blue}{\left(b \cdot i\right)} \cdot t \]

   10. Applied egg-rr 41.5%

       \[\leadsto \color{blue}{\left(b \cdot i\right) \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -1.08 \cdot 10^{+40}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;i \leq 1.26 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(b \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;i \leq 1.26 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -2.8e+39)
   (* b (* t i))
   (if (<= i 1.26e-179) (* x (* y z)) (* i (* t b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -2.8e+39) {
		tmp = b * (t * i);
	} else if (i <= 1.26e-179) {
		tmp = x * (y * z);
	} else {
		tmp = i * (t * b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (i <= (-2.8d+39)) then
        tmp = b * (t * i)
    else if (i <= 1.26d-179) then
        tmp = x * (y * z)
    else
        tmp = i * (t * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -2.8e+39) {
		tmp = b * (t * i);
	} else if (i <= 1.26e-179) {
		tmp = x * (y * z);
	} else {
		tmp = i * (t * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if i <= -2.8e+39:
		tmp = b * (t * i)
	elif i <= 1.26e-179:
		tmp = x * (y * z)
	else:
		tmp = i * (t * b)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -2.8e+39)
		tmp = Float64(b * Float64(t * i));
	elseif (i <= 1.26e-179)
		tmp = Float64(x * Float64(y * z));
	else
		tmp = Float64(i * Float64(t * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (i <= -2.8e+39)
		tmp = b * (t * i);
	elseif (i <= 1.26e-179)
		tmp = x * (y * z);
	else
		tmp = i * (t * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -2.8e+39], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 1.26e-179], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if i < -2.80000000000000001e39

   1. Initial program 63.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 63.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 63.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

      3. *-lowering-*.f64 43.1

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

   8. Simplified 43.1%

      \[\leadsto \color{blue}{b \cdot \left(t \cdot i\right)} \]

   if -2.80000000000000001e39 < i < 1.2599999999999999e-179

   1. Initial program 79.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(a \cdot t\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \left(\color{blue}{a \cdot \left(\mathsf{neg}\left(t\right)\right)} + y \cdot z\right) \]

      5. mul-1-neg N/A

         \[\leadsto x \cdot \left(a \cdot \color{blue}{\left(-1 \cdot t\right)} + y \cdot z\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot t, y \cdot z\right)} \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(t\right)}, y \cdot z\right) \]

      9. *-lowering-*.f64 56.2

         \[\leadsto x \cdot \mathsf{fma}\left(a, -t, \color{blue}{y \cdot z}\right) \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(a, -t, y \cdot z\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]

      2. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

      3. *-lowering-*.f64 35.6

         \[\leadsto x \cdot \color{blue}{\left(z \cdot y\right)} \]

   8. Simplified 35.6%

      \[\leadsto \color{blue}{x \cdot \left(z \cdot y\right)} \]

   if 1.2599999999999999e-179 < i

   1. Initial program 64.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 65.7

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 65.7%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 41.5

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Simplified 41.5%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{elif}\;i \leq 1.26 \cdot 10^{-179}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(c \cdot j\right)\\ \mathbf{if}\;j \leq -0.000195:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 7 \cdot 10^{+57}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* c j))))
   (if (<= j -0.000195) t_1 (if (<= j 7e+57) (* i (* t b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (c * j);
	double tmp;
	if (j <= -0.000195) {
		tmp = t_1;
	} else if (j <= 7e+57) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (c * j)
    if (j <= (-0.000195d0)) then
        tmp = t_1
    else if (j <= 7d+57) then
        tmp = i * (t * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (c * j);
	double tmp;
	if (j <= -0.000195) {
		tmp = t_1;
	} else if (j <= 7e+57) {
		tmp = i * (t * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (c * j)
	tmp = 0
	if j <= -0.000195:
		tmp = t_1
	elif j <= 7e+57:
		tmp = i * (t * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(c * j))
	tmp = 0.0
	if (j <= -0.000195)
		tmp = t_1;
	elseif (j <= 7e+57)
		tmp = Float64(i * Float64(t * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (c * j);
	tmp = 0.0;
	if (j <= -0.000195)
		tmp = t_1;
	elseif (j <= 7e+57)
		tmp = i * (t * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -0.000195], t$95$1, If[LessEqual[j, 7e+57], N[(i * N[(t * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -1.94999999999999996e-4 or 6.9999999999999995e57 < j

   1. Initial program 66.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 46.8

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 46.8%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

      2. *-lowering-*.f64 38.9

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

   8. Simplified 38.9%

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

   if -1.94999999999999996e-4 < j < 6.9999999999999995e57

   1. Initial program 72.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 46.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 46.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 36.5

         \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]

   8. Simplified 36.5%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -0.000195:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{elif}\;j \leq 7 \cdot 10^{+57}:\\ \;\;\;\;i \cdot \left(t \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(c \cdot j\right)\\ \mathbf{if}\;j \leq -0.015:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 1.75 \cdot 10^{+58}:\\ \;\;\;\;b \cdot \left(t \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (* c j))))
   (if (<= j -0.015) t_1 (if (<= j 1.75e+58) (* b (* t i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (c * j);
	double tmp;
	if (j <= -0.015) {
		tmp = t_1;
	} else if (j <= 1.75e+58) {
		tmp = b * (t * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * (c * j)
    if (j <= (-0.015d0)) then
        tmp = t_1
    else if (j <= 1.75d+58) then
        tmp = b * (t * i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * (c * j);
	double tmp;
	if (j <= -0.015) {
		tmp = t_1;
	} else if (j <= 1.75e+58) {
		tmp = b * (t * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = a * (c * j)
	tmp = 0
	if j <= -0.015:
		tmp = t_1
	elif j <= 1.75e+58:
		tmp = b * (t * i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * Float64(c * j))
	tmp = 0.0
	if (j <= -0.015)
		tmp = t_1;
	elseif (j <= 1.75e+58)
		tmp = Float64(b * Float64(t * i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = a * (c * j);
	tmp = 0.0;
	if (j <= -0.015)
		tmp = t_1;
	elseif (j <= 1.75e+58)
		tmp = b * (t * i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -0.015], t$95$1, If[LessEqual[j, 1.75e+58], N[(b * N[(t * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -0.014999999999999999 or 1.7499999999999999e58 < j

   1. Initial program 66.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

      2. +-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      10. neg-lowering-neg.f64 46.8

          \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

   5. Simplified 46.8%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

      2. *-lowering-*.f64 38.9

         \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

   8. Simplified 38.9%

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

   if -0.014999999999999999 < j < 1.7499999999999999e58

   1. Initial program 72.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(b \cdot t\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{1} \cdot \left(b \cdot t\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{b \cdot t}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, b \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, b \cdot t\right) \]

      11. *-lowering-*.f64 46.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, -y, \color{blue}{b \cdot t}\right) \]

   5. Simplified 46.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, -y, b \cdot t\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

      3. *-lowering-*.f64 35.9

         \[\leadsto b \cdot \color{blue}{\left(t \cdot i\right)} \]

   8. Simplified 35.9%

      \[\leadsto \color{blue}{b \cdot \left(t \cdot i\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 22.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(c \cdot j\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* c j)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (c * j);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = a * (c * j)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (c * j);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (c * j)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(c * j))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (c * j);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Add Preprocessing

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]

   2. +-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j + -1 \cdot \left(t \cdot x\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto a \cdot \left(\color{blue}{j \cdot c} + -1 \cdot \left(t \cdot x\right)\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(j, c, -1 \cdot \left(t \cdot x\right)\right)} \]

   5. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{\mathsf{neg}\left(t \cdot x\right)}\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)}\right) \]

   7. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, \color{blue}{t \cdot \left(-1 \cdot x\right)}\right) \]

   9. mul-1-neg N/A

      \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

   10. neg-lowering-neg.f64 36.9

       \[\leadsto a \cdot \mathsf{fma}\left(j, c, t \cdot \color{blue}{\left(-x\right)}\right) \]

5. Simplified 36.9%

   \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(j, c, t \cdot \left(-x\right)\right)} \]

6. Taylor expanded in j around inf

   \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]

   2. *-lowering-*.f64 22.1

      \[\leadsto a \cdot \color{blue}{\left(c \cdot j\right)} \]

8. Simplified 22.1%

   \[\leadsto \color{blue}{a \cdot \left(c \cdot j\right)} \]
9. Add Preprocessing

Developer Target 1: 61.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_2 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + t\_1\\ \mathbf{if}\;x < -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x < 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i))))
        (t_2
         (+
          (-
           (* x (- (* y z) (* t a)))
           (/
            (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0)))
            (+ (* c z) (* t i))))
          t_1)))
   (if (< x -1.469694296777705e-64)
     t_2
     (if (< x 3.2113527362226803e-147)
       (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) t_1))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (pow((c * z), 2.0) - pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = j * ((c * a) - (y * i))
    t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ** 2.0d0) - ((t * i) ** 2.0d0))) / ((c * z) + (t * i)))) + t_1
    if (x < (-1.469694296777705d-64)) then
        tmp = t_2
    else if (x < 3.2113527362226803d-147) then
        tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double t_2 = ((x * ((y * z) - (t * a))) - ((b * (Math.pow((c * z), 2.0) - Math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1;
	double tmp;
	if (x < -1.469694296777705e-64) {
		tmp = t_2;
	} else if (x < 3.2113527362226803e-147) {
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((c * a) - (y * i))
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (math.pow((c * z), 2.0) - math.pow((t * i), 2.0))) / ((c * z) + (t * i)))) + t_1
	tmp = 0
	if x < -1.469694296777705e-64:
		tmp = t_2
	elif x < 3.2113527362226803e-147:
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_2 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(Float64(b * Float64((Float64(c * z) ^ 2.0) - (Float64(t * i) ^ 2.0))) / Float64(Float64(c * z) + Float64(t * i)))) + t_1)
	tmp = 0.0
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = Float64(Float64(Float64(Float64(b * i) - Float64(x * a)) * t) - Float64(Float64(z * Float64(c * b)) - t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * ((c * a) - (y * i));
	t_2 = ((x * ((y * z) - (t * a))) - ((b * (((c * z) ^ 2.0) - ((t * i) ^ 2.0))) / ((c * z) + (t * i)))) + t_1;
	tmp = 0.0;
	if (x < -1.469694296777705e-64)
		tmp = t_2;
	elseif (x < 3.2113527362226803e-147)
		tmp = (((b * i) - (x * a)) * t) - ((z * (c * b)) - t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[Power[N[(c * z), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(t * i), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * z), $MachinePrecision] + N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[Less[x, -1.469694296777705e-64], t$95$2, If[Less[x, 3.2113527362226803e-147], N[(N[(N[(N[(b * i), $MachinePrecision] - N[(x * a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] - N[(N[(z * N[(c * b), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -293938859355541/2000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 32113527362226803/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))